Favorite sites of randomly biased walks on a supercritical Galton–Watson tree

Abstract

Erdős and Révész (1984) initiated the study of favorite sites by considering the one-dimensional simple random walk. We investigate in this paper the same problem for a class of null-recurrent randomly biased walks on a supercritical Galton–Watson tree. We prove that there is some parameter κ∈(1,∞] such that the set of the favorite sites of the biased walk is almost surely bounded in the case κ∈(2,∞], tight in the case κ=2, and oscillates between a neighborhood of the root and the boundary of the range in the case κ∈(1,2). Moreover, our results yield a complete answer to the cardinality of the set of favorite sites in the case κ∈(2,∞]. The proof relies on the exploration of the Markov property of the local times process with respect to the space variable and on a precise tail estimate on the maximum of local times, using a change of measure for multi-type Galton–Watson trees.